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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 02}
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%\date{2021 年 2 月 28 日}
\date{March 16, 2021}
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%\subsection{Homework 02}
%E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.4. 

\begin{document}

\maketitle

\begin{enumerate}
\item [E3.4.1.] Find the mean time to reach state 3 starting from state 0 for the Markov chain whose transition probability matrix is
\begin{eqnarray*}
P=
\begin{blockarray}{ccccc}
& 0 & 1 & 2 & 3 \\
\begin{block}{c[cccc]}
  0 & 0.4 & 0.3 & 0.2 & 0.1 \\
  1 & 0    & 0.7 & 0.2 & 0.1 \\ 
  2 & 0    & 0   &  0.9 & 0.1 \\
  3 & 0    & 0   &  0    &1 \\
\end{block}
\end{blockarray}.
\end{eqnarray*}


\item [E3.4.2.] Consider the Markov chain whose transition probability matrix is given by 
\begin{eqnarray*}
P=
\begin{blockarray}{cccc}
& 0 & 1 & 2 \\
\begin{block}{c[ccc]}
  0 & 1    & 0    & 0 \\
  1 & 0.1 & 0.6 & 0.3 \\ 
  2 & 0    & 0   &  1 \\
\end{block}
\end{blockarray}.
\end{eqnarray*}
\begin{enumerate}
\item  Starting in state 1, determine the probability that the Markov chain ends in state 0.
\item  Determine the mean time to absorption.
\end{enumerate}


\item [P3.4.1.] Which will take fewer flips, on average: successively flipping a quarter until the pattern HHT appears, i.e., until you observe two successive heads followed by a tails; or successively flipping a quarter until the pattern HTH appears? Can you explain why these are different?


\item [P3.4.5.] A white rat is put into compartment 4 of the maze shown here. It moves through the compartments at random; i.e., if there are $k$ ways to leave a compartment, it chooses each of these with probability $1/k$. What is the probability that it finds the food in compartment 3 before feeling the electric shock in compartment 7?

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\item [E3.5.1.] The probability of the thrower winning in the dice game called `craps' is $p = 0.4929$. Suppose Player A is the thrower and begins the game with \$5, and Player B, his opponent, begins with \$10. What is the probability that Player A goes bankrupt before Player B? Assume that the bet is \$1 per round.


\item [P3.5.4.] Martha has a fair die with the usual six sides. She throws the die and records the number. She throws the die again and adds the second number to the first. She repeats this until the cumulative sum of all the tosses first exceeds 10. What is the probability that she stops at a cumulative sum of 13?


\end{enumerate}


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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.4. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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\begin{enumerate}
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